A finite group $G$ is an $n$-transposition group if there exists a union $D\subset G$ of conjugacy classes of involutions such that $\langle D \rangle = G$ and for all $a,b\in D$, the product $ab$ is of order at most $n$.

The almost simple $3$-transposition groups were classified by Bernd Fischer. Among the groups classified are the three sporadic simple Fischer groups. It is also well-known that the Baby Monster group is a $4$-transposition group and the Monster is a $6$-transposition group.

Have $n$-transposition groups been classified or investigated for any other $n$ (especially $n=4,5,6$)?

What are the values of $n$ for the other exceptional and sporadic simple groups?

Are there other important generalisations of the $3$-tranposition property? (From Aschbacher's book on $3$-transposition groups, I see that there is a generalisation due to Timmesfeld to a $\{3,4\}^+$-transposition property, which is related to root elements of groups of Lie type.)